f(x) = x^2 - 8x + 19 (a) Express f(x) in the form (x + a)^2 + b, where a and b are constants - Edexcel - A-Level Maths Pure - Question 7 - 2017 - Paper 1

To sketch the graph of the function:

1. Identify the vertex of the parabola which represents the minimum point Q:

   • The vertex form given is (x - 4)^2 + 3, indicating that Q is at (4, 3).

2. Find the point P where the curve crosses the y-axis (x = 0):

   • Substitute x = 0 into f(x): $f(0) = (0 - 4)^2 + 3 = 16 + 3 = 19$
   • Therefore, P is at (0, 19).

3. The graph should show:

   • A U-shaped curve opening upwards.
   • Coordinates P at (0, 19) and Q at (4, 3) should both be marked clearly.

To find the distance PQ, we can use the distance formula between points P(0, 19) and Q(4, 3):

1. Apply the distance formula: $PQ = \\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

   • Where (x_1, y_1) = (0, 19) and (x_2, y_2) = (4, 3).

2. Substitute the coordinates into the formula: $PQ = \\sqrt{(4 - 0)^2 + (3 - 19)^2}$

3. Calculate: $PQ = \\sqrt{4^2 + (-16)^2} = \\sqrt{16 + 256} = \\sqrt{272}$

4. Simplify the surd: $PQ = \\sqrt{16 imes 17} = 4\\sqrt{17}$

So, the simplified distance PQ is: $PQ = 4\\sqrt{17}$.